A065478 -id:A065478 - cp's OEIS frontend

A078079 Continued fraction expansion of Product_{p prime}(1 - p/(p^3-1)).

0, 1, 1, 2, 1, 3, 1, 3, 1, 2, 1, 77, 2, 1, 1, 10, 2, 1, 1, 1, 7, 1, 1, 3, 3, 5, 5, 3, 1, 1, 35, 1, 8, 3, 1, 2, 1, 3, 2, 5, 3, 1, 14, 1, 2, 1, 15, 1, 8, 1, 2, 3, 6, 1, 3, 1, 7, 1, 1, 2, 1, 1, 24, 1, 2, 3, 14, 23, 1, 9, 1, 46, 7, 1, 1, 5, 1, 1, 14, 2, 1, 20, 2, 1, 6, 1, 1, 2, 2, 1, 1, 1, 35, 1

Offset: 0

Benoit Cloitre, Dec 02 2002

Emanuele Munarini and Norma Zagaglia Salvi, Binary strings without zigzags, Séminaire Lotharingien de Combinatoire, B49h (2004), 15 pp.

Cf. A065478 (decimal expansion).

contfrac(prodeulerrat(1 - p/(p^3-1))) \\ Amiram Eldar, Mar 17 2021

Offset changed by Andrew Howroyd, Jul 05 2024

A351435 If n = Product (p_j^k_j) then a(n) = Product ((p_j + 1)^(k_j + 1)).

Original entry on oeis.org

1, 9, 16, 27, 36, 144, 64, 81, 64, 324, 144, 432, 196, 576, 576, 243, 324, 576, 400, 972, 1024, 1296, 576, 1296, 216, 1764, 256, 1728, 900, 5184, 1024, 729, 2304, 2916, 2304, 1728, 1444, 3600, 3136, 2916, 1764, 9216, 1936, 3888, 2304, 5184, 2304, 3888, 512, 1944, 5184, 5292, 2916

Offset: 1

Ilya Gutkovskiy, Feb 11 2022

Cf. A003557, A003958, A003959, A048250, A064549, A065478, A306633, A326297, A327564, A351434.

a:= n-> mul((i[1]+1)^(i[2]+1), i=ifactors(n)[2]):
seq(a(n), n=1..53);  # Alois P. Heinz, Feb 11 2022

f[p_, e_] := (p + 1)^(e + 1); a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Table[a[n], {n, 1, 53}]

a(n) = my(f=factor(n)); for (k=1, #f~, f[k,1]++; f[k,2]++); factorback(f); \\ Michel Marcus, Feb 11 2022

a(n) = A003959(n) * A048250(n).
Sum_{k=1..n} a(k) ~ c * n^3, where c = 1/(3 * Product_{p prime} (1 - p/(p^3-1))) = 1 /(3 * A065478) = 0.5787439255... . - Amiram Eldar, Nov 19 2022
Sum_{n>=1} 1/a(n) = zeta(2)/zeta(3) (A306633). - Amiram Eldar, Dec 15 2023

cp's OEIS Frontend

A078079 Continued fraction expansion of Product_{p prime}(1 - p/(p^3-1)).

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